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An implicit Euler finite-volume scheme for an $n$-species populationcross-diffusion system of Shigesada--Kawasaki--Teramoto-type in a boundeddomain with no-flux boundary conditions is proposed and analyzed. The schemepreserves the formal gradient-flow or entropy structure and preserves thenonnegativity of the population densities. The key idea is to consider asuitable mean of the mobilities in such a way that a discrete chain rule isfulfilled and a discrete analog of the entropy inequality holds. The existenceof finite-volume solutions, the convergence of the scheme, and the large-timeasymptotics to the constant steady state are proven. Furthermore, numericalexperiments in one and two space dimensiona for two and three species arepresented. The results are valid for a more general class of cross-diffusionsystems satisfying some structural conditions.

A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system